How Do You Ensure a Positive Z-Component in a Normal Vector?

[seanuleh]

Homework Statement

Find the unit vector with positive z component which is normal to the surface z=xy+xy^2 at the point (1,1,2) on the surface.

Homework Equations

Well no real relevant equations i guess... but the thing is i know how to get the normal vector, i know how to convert it to a unit vector i just have no idea how to make the z-component positive. A friend suggested that i just change the z-component to 1 in the solution however i am convinced that this is an entirely different vector and is more than likely NOT the solution. Please help :(

The Attempt at a Solution

F_x = y + y^2
F_x(1,1,2) = 2
F_y = x + 2xy
F_y(1,1,2) = 3

so the normal vector would be [2, 3, -1].
Please help :(

 

[Defennder]

What do you know about the gradient vector? Another way to do it, without using that is to first parametrise the surface and use some vector manipulation to get the normal vector from there. Anyway your answer appears to be correct.

 

[seanuleh]

while my answer is a valid normal vector, it does not have a positive z-component so it is not actually correct, my question is how do i go about making it have a positive z-component.

 

[HallsofIvy]

Multiplying the vector by -1, to get [-2, -3, 1], gives a vector that points in exactly the opposite direction and so is still perpendicular to the surface but has z-component positive.

 

[D H]

What makes you think the z component has to be positive? Think about it this way. Consider the surface z=0 (i.e., the x-y plane). The vector [noparse][0,0,1][/noparse] is normal to the surface, but so is the vector [noparse][0,0,-1][/noparse], and so is any vector of the form [noparse][0,0,a][/noparse] where a is any real number.

There is a big hint in the above. Another hint: Any vector parallel to a normal vector is also a normal vector. How can you make a vector that is parallel to the one you constructed that does have the desired characteristics of being a unit vector and having a positive z component?

 

[HallsofIvy]

What makes you think the z component has to be positive? Think about it this way. Consider the surface z=0 (i.e., the x-y plane). The vector [noparse][0,0,1][/noparse] is normal to the surface, but so is the vector [noparse][0,0,-1][/noparse], and so is any vector of the form [noparse][0,0,a][/noparse] where a is any real number.

The problem asked for a unit vector, normal to the surface, with positive z component!

There is a big hint in the above. Another hint: Any vector parallel to a normal vector is also a normal vector. How can you make a vector that is parallel to the one you constructed that does have the desired characteristics of being a unit vector and having a positive z component?

Now that's a good hint!